Finite Group Databasebeta

← #6519705

Subgroups — #6519705

Order 512.

2096 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3215C211yesmaximal, characteristic
2C2×#3104C211yesmaximal, characteristic, complemented
3C2×#2545C211yesmaximal, characteristic
4#34235C211yesmaximal, characteristic, complemented
5#28123C212yesmaximal
6#28121C212yesmaximal, complemented
7#25379C214yesmaximal, complemented
8#25286C214yesmaximal, complemented
9#24736C214yesmaximal, complemented
10#24637C214yesmaximal, complemented
11#24217C212yesmaximal
12#24212C212yesmaximal, complemented
13#21953C211yesmaximal, characteristic, complemented
14#21806C211yesmaximal, characteristic
15#20741C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21#3215—28no—
22C2×#497C2212yes—
23C2×#489C2212yes—
24#3104—28no—
25C2×#484C2212yes—
26C2×#477C2212yescomplemented
27C2×#464C2214yes—
28C2×#464C2214yes—
29C2×#462C2212yescomplemented
30C2×#462C2214yescomplemented
31C2×#451C2212yes—
32C2×#451C2212yes—
33C2×#450C2212yes—
34C2×C4×Q16C2214yescomplemented
35C2×#432C2211yescharacteristic
36C2×#431C2211yescharacteristic
37#2545—21no—
38#2545—21no—
39#2545—21no—
40#2545—21no—
41#2545C2214yes—
42#2545C2214yes—
43C2×#396—22no—
44#2365C2212yescomplemented
45#2365C2212yescomplemented
46#2360—22no—
47#2356C2212yescomplemented
48#2351C2212yescomplemented
49#2343—22no—
50#2336—21no—
51#2336—21no—
52#2320—21no—
53C4×C22⋊Q8C2212yes—
54C4×C22⋊Q8C2212yes—
55C2×#391C2211yescharacteristic
56C2×#390C2211yescharacteristic
57C2×#388—22no—
58#2283C2211yescharacteristic
59#2276—21no—
60C2×C4×C4⋊C4C2211yescharacteristic
61#2068C2214yescomplemented
62#2067C2214yes—
63#2056C2214yescomplemented
64#2055C2214yes—
65#2043C2214yescomplemented
66#2042C2214yes—
67#2016C2214yescomplemented
68#2015C2214yes—
69#1983C2214yes—
70#1983C2214yes—
71#1981C2214yescomplemented
72#1958C2212yes—
73#1958C2212yes—
74#1940C2214yescomplemented
75#1939C2214yescomplemented
76#1928C2214yescomplemented
77#1925C2214yes—
78#1880C2214yes—
79#1878C2214yescomplemented
80#1824C2212yes—
81#1805C2212yescomplemented
82#1805C2212yescomplemented
83#1797C2212yes—
84#1797C2212yes—
85#1787C2212yes—
86#1787C2212yes—
87#1786C2212yes—
88#1763C2214yescomplemented
89C23×Q8C2311yescharacteristic
90C23×Q8C2311yescharacteristic
91C23×D4C2311yescharacteristic
92C22×Q16C2314yes—
93C22×Q16C2314yes—
94C22×SD16C2314yes—
95#533—24no—
96#533D418yes—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—44no—
100C2×C4⋊Q8C2314yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×#125—21no—
104C2×#124—21no—
105C2×#124—21no—
106C2×#124—21no—
107C2×#124—21no—
108C2×#124—24no—
109C2×#124—24no—
110C2×#124—44no—
111C2×#124C2312yes—
112C2×#124C2312yes—
113C2×#124C2314yes—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8C2312yes—
116C2×C4×Q8—22no—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8—24no—
121C2×C4×Q8—24no—
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2312yes—
127C2×#117—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4D412yes—
131C22×C4⋊C4C2311yescharacteristic
132C22×C22⋊C4—21no—
133C22×C22⋊C4—22no—
134C22×C22⋊C4—22no—
135C22×C22⋊C4—21no—
136C22×C42—21no—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140#497—28no—
141#489—28no—
142#484—28no—
143#477—28no—
144#464—216no—
145#464—48no—
146#462—28no—
147#462—48no—
148#451—28no—
149#451—28no—
150#450—28no—
151C4×Q16—216no—
152#432—24no—
153#431—24no—
154C2×#107C2314yes—
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2312yes—
157C2×C4⋊C8C2314yes—
158C2×C4⋊C8C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#102C2314yes—
167C2×#102C2314yes—
168C2×#102C2314yes—
169C2×C22⋊C8C2312yes—
170C2×C22⋊C8C2312yes—
171C2×#97C2312yes—
172C2×C4×C8C2312yes—
173#396—24no—
174#396—24no—
175#396—24no—
176#396—24no—
177#391—22no—
178#391—21no—
179#391—21no—
180#391C2312yes—
181#391C2312yes—
182#391C2314yes—
183#390—21no—
184#390—21no—
185#390C2314yes—
186#389—22no—
187#389—22no—
188#389—22no—
189#389C2312yes—
190#389C2312yes—
191#388—24no—
192#388—24no—
193#388—24no—
194#388—24no—
195#387—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#385—22no—
204#385—22no—
205#383—21no—
206#383—22no—
207#382—22no—
208#382C2311yescharacteristic
209#382C2311yescharacteristic
210#381—21no—
211#380C2312yes—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—21no—
228#376—21no—
229#376—22no—
230C2×#95—22no—
231C2×#95D412yes—
232#340C2314yes—
233#340C2314yes—
234#336C2314yes—
235#336C2314yes—
236C25—21no—
237C22×Q8—24no—
238C22×Q8—21no—
239C22×Q8—21no—
240C22×Q8—21no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—42no—
246C22×Q8—42no—
247C22×Q8—42no—
248C22×Q8C2412yes—
249C22×Q8C2412yes—
250C22×Q8C2412yes—
251C22×D4—22no—
252C22×D4—21no—
253C22×D4—22no—
254C22×D4—42no—
255C22×D4—42no—
256C22×D4C2412yes—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4C2×D412yes—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D412yes—
263C23×C4C2×D411yescharacteristic
264C23×C4C2411yescharacteristic
265C2×Q16—28no—
266C2×Q16—28no—
267C2×Q16—28no—
268C2×Q16—44no—
269C2×Q16—44no—
270C2×Q16—44no—
271C2×SD16—28no—
272C2×SD16—44no—
273C2×SD16—44no—
274C22×C8C2414yes—
275C22×C8C2414yes—
276C4⋊Q8—28no—
277C4⋊Q8—22no—
278C4⋊Q8—22no—
279C4⋊Q8—44no—
280C4⋊Q8—44no—
281C4⋊Q8—44no—
282C4⋊Q8C2×D414yes—
283C4⋊Q8C2×D414yes—
284#125—28no—
285#124—28no—
286#124—24no—
287#124—24no—
288#124—24no—
289#124—24no—
290#124—44no—
291#124—44no—
292#124—44no—
293#124—44no—
294#124—44no—
295C22⋊Q8—28no—
296C22⋊Q8—24no—
297C22⋊Q8C2×D418yes—
298C4×Q8—28no—
299C4×Q8—24no—
300C4×Q8—24no—
301C4×Q8—24no—
302C4×Q8—24no—
303C4×Q8—24no—
304C4×Q8—44no—
305C4×Q8—44no—
306C4×Q8—48no—
307C4×Q8—44no—
308C4×Q8—44no—
309C4×Q8C2×D418yes—
310#117—28no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—48no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—48no—
349C2×C22⋊C4—48no—
350C2×C22⋊C4—44no—
351C2×C22⋊C4—44no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—42no—
361C2×C42—42no—
362C2×C42—44no—
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D414yes—
370C2×C42#4114yes—
371C2×C42C2412yes—
372C2×C42C2412yes—
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375#107—28no—
376C4⋊C8—24no—
377C4⋊C8—24no—
378C4⋊C8—44no—
379C4⋊C8—44no—
380#103—28no—
381#103—28no—
382#103—28no—
383#103—28no—
384#103—44no—
385#103—44no—
386#103—44no—
387#102—28no—
388#102—44no—
389#102—44no—
390C22⋊C8—24no—
391C22⋊C8—24no—
392#97—24no—
393C4×C8—24no—
394#95—24no—
395#95—24no—
396#95—21no—
397#95—22no—
398#95—21no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—21no—
404#95—21no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95C2×D412yes—
411#95C2×D412yes—
412#95C2×D412yes—
413#95C2×D412yes—
414C24—22no—
415C24—42no—
416C24—44no—
417C24—42no—
418C24—42no—
419C24—44no—
420C24C22×D411yescharacteristic
421C2×Q8—24no—
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—24no—
430C2×Q8—24no—
431C2×Q8—22no—
432C2×Q8—24no—
433C2×Q8—22no—
434C2×Q8—24no—
435C2×Q8—48no—
436C2×Q8—48no—
437C2×Q8—44no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—84no—
444C2×Q8—84no—
445C2×Q8C22×D414yes—
446C2×Q8H(5,𝔽2)14yes—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—24no—
450C2×D4—24no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—84no—
455C2×D4—84no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—44no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4#12114yes—
482C22×C4#12114yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C2×#4112yes—
494C22×C4C2511yescharacteristic
495Q16—416no—
496Q16—88no—
497SD16—88no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8—24no—
502C2×C8—24no—
503C2×C8—24no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—28no—
508C4⋊C4—28no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—48no—
519C4⋊C4—48no—
520C4⋊C4—44no—
521C4⋊C4—42no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C4⋊C4C22×D414yes—
528C22⋊C4—28no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—48no—
538C42—28no—
539C42—24no—
540C42—24no—
541C42—41no—
542C42—41no—
543C42—41no—
544C42—41no—
545C42—44no—
546C42—44no—
547C42—44no—
548C42—44no—
549C42—44no—
550C42—82no—
551C42—82no—
552C42#12114yes—
553C42#12114yes—
554C42C22×D412yes—
555C42C22×D412yes—
556C42C2×#4112yes—
557C42C2×#4112yes—
558C23—21no—
559C23—21no—
560C23—22no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—44no—
565C23—44no—
566C23—84no—
567C23—88no—
568C23—88no—
569C23—84no—
570C23—84no—
571C23C2×#12112yes—
572C23#57814yes—
573C23C23×D411yescharacteristic
574Q8—28no—
575Q8—24no—
576Q8—24no—
577Q8—48no—
578Q8—48no—
579Q8—42no—
580Q8—44no—
581Q8—42no—
582Q8—44no—
583Q8—44no—
584Q8—84no—
585D4—44no—
586D4—44no—
587D4—84no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—28no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—44no—
612C2×C4—48no—
613C2×C4—48no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4C2×#12112yes—
635C2×C4C2×#12112yes—
636C2×C4C2×#12112yes—
637C2×C4C2×#12112yes—
638C2×C4C2×#12112yes—
639C2×C4#57514yes—
640C2×C4C23×D411yescharacteristic
641C2×C4C22×#4111yescharacteristic
642C2×C4C2×H(5,𝔽2)12yes—
643C2×C4#58512yes—
644C8—44no—
645C8—44no—
646C22—24no—
647C22—24no—
648C22—22no—
649C22—22no—
650C22—82no—
651C22—84no—
652C22—84no—
653C22—82no—
654C22—82no—
655C22—168no—
656C22#304414yes—
657C22C22×#12111yescharacteristic
658C22#348411yescharacteristic
659C22#348711yescharacteristic
660C22C2×#57512yes—
661C22C2×#57812yes—
662C4—22no—
663C4—22no—
664C4—24no—
665C4—22no—
666C4—22no—
667C4—22no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—44no—
672C4—44no—
673C4—44no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4#348812yes—
679C4#349412yes—
680C2—24no—
681C2—162no—
682C2#3556311yescharacteristic
683C2#6041511yescharacteristic
684C2C2×#304411yescharacteristic
685C2#6160312yes—
686C2#6190512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#3104C2semidirect
4. #34235C2semidirect
6. #28121C2semidirect
7. #25379C2semidirect
8. #25286C2semidirect
9. #24736C2semidirect
10. #24637C2semidirect
12. #24212C2semidirect
13. #21953C2semidirect
15. #20741C2semidirect
26. C2×#477C22semidirect
29. C2×#462C22semidirect
30. C2×#462C22semidirect
34. C2×C4×Q16C22semidirect
44. #2365C22semidirect
45. #2365C22semidirect
47. #2356C22semidirect
48. #2351C22semidirect
61. #2068C22semidirect
63. #2056C22semidirect
65. #2043C22semidirect
67. #2016C22semidirect
71. #1981C22semidirect
74. #1940C22semidirect
75. #1939C22semidirect
76. #1928C22semidirect
79. #1878C22semidirect
81. #1805C22semidirect
82. #1805C22semidirect
88. #1763C22semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#32152
C2×#31042
C2×#25452
#342352
#281232
#281212
#253792
#252862
#247362
#246372
#242172
#242122
#219532
#218062
#207412